broken screen and not turning on

charley, if you have not replaced the battery and if you have owned the ipod for a while, my suggestion would be to definitely change the battery. I am certain that it is at the end of its lifespan. Personally, I would go ahead and replace the screen. Those are all relatively easy repairs and once you have a new battery and screen you can further diagnose if needed. your parts are available here http://www.ifixit.com/iPod-Parts/iPod-3r... and I am sure other places as well. As for what else could be wrong with it would strictly be a guessing game until you replaced those parts. Good Luck and let us know how it is going.